Research ArticleOPTICS

Optical coherence transfer mediated by free electrons

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Science Advances  30 Apr 2021:
Vol. 7, no. 18, eabf6380
DOI: 10.1126/sciadv.abf6380

Figures

  • Fig. 1 Linear and nonlinear Mach-Zehnder interferometer incorporating a free-electron beam.

    (A) Proposed experimental concept based on an electron beam (e-beam, illustrated in green) in a transmission electron microscope. A laser field imprints optical-phase information on the electron, which, after propagation, can transfer it back to the radiation via CL emission, typically near the sample section of the microscope. (B) Scheme for a Mach-Zehnder interferometer with a reference laser and optical coherence carried by a free-electron beam. Intensity correlations between the two interferometer ports, measured as I1 and I2, are used to retrieve information on the electron arm of the interferometer, as well as on the sample. (C) Nonlinear Mach-Zehnder interferometry can reveal information on high-order components of the electron state and the CL by interfering with harmonic frequencies of the reference field.

  • Fig. 2 Coherent properties of CL emission by shaped electrons.

    (A) (Top) Laser-driven PINEM imprints phase oscillations on an electron wave packet, regardless of the exact arrival timing of the electron. The electron-phase oscillations are equivalent to a temporal lens, slowing and accelerating parts of the wave packet periodically. (Bottom) After propagation, anharmonic density and phase modulations evolve in the electron, leading to coherent CL (e.g., in a waveguide). (B) Calculated evolution of the degree of coherence (DOC) following PINEM as a function of the propagation distance z. The DOC is nonzero only for harmonics of the PINEM-driving laser. The cutoff of the harmonic DOC spectrum is determined by the span of the electron-energy ladder population. For PINEM-modulated electron, the cutoff harmonic order is cutoffDOC ≈ 4∣β∣ = 16. (C) The emitted field is proportional to DOC, and its width is inversely proportional to the pulse duration of the pre-structured electron density (see legend for FWHM electron pulse durations). While the DOC and its square root comprise a harmonic frequency comb that varies with z, the emission rate of CL photons (red dashed line) remains unaffected by the temporal structure and only depends on the coupling amplitude, gω, between the electron and the optical modes. The panels in (C) refer to z = 6.43 mm [white dashed line in (B)].

  • Fig. 3 Correlations in the CL into a dielectric waveguide produced by modulated electrons.

    (A) Conceptual scheme of the experimental setting on the basis of photonic circuitry. A laser field structures a pulsed electron beam, which excites coherent radiation in a parallel-aligned waveguide. Mixing a replica of the driving laser field, ER, with the CL field, ECL, results in a difference between the signals recorded in the detectors at the output ports of the interferometer, I1 and I2. (B) Properties of CL emitted into the waveguide for an interaction length of 30 μm positioned at a distance z = 6.43 mm away from the electron modulation region. The coherent CL amplitude (blue) is the product of the spectral coupling amplitude, ∣gω∣ (red), and DOC(ω) (gray), where phase matching between the electron and the guided modes limits the CL emission to a single harmonic peak. (C) to (H) show CL properties as a function of the propagation length near the fiber, Lprop: (C) to (E) show the spectral distribution of ∣gω∣ and âω, whereas (F) to (H) show the emitted CL-field pulse in the time domain, E(t). The different columns show the CL properties for an electron propagation length Lprop = 10 μm (C and F), 100 μm (D and G), and 1 mm (E and H) along the waveguide. The effect of Lprop on the peak probability of emission is reflected in the vertical scale. When the coupling bandwidth is narrower than the width of the DOC [e.g., in (E)], the CL pulse is temporally distorted (H). Details of the coupling into a cylindrical waveguide are presented elsewhere (36).

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